Non-Supersymmetric Attractors in $R^2$ Gravities

TL;DR

The work demonstrates that non-supersymmetric attractor behavior survives in higher-derivative gravity, showcasing analytic near-horizon solutions in general $R^2$ gravity and extending to Gauss–Bonnet couplings with moduli. The horizon values of moduli are fixed by extremizing a horizon function $W_H$ (with $W_H=V-2(eta+2 ext{del} s)G$ in $R^2$ gravity and $W_H=V+4G$ in Gauss–Bonnet theory), and regularity of the scalar fields at the horizon guarantees attractor behavior even when backreaction and higher-derivative terms are included. The analysis employs Frobenius-type expansions near the horizon and, in the Gauss–Bonnet case, perturbative $oldsymbol{\alpha}$-expansion to illuminate both analytic and non-analytic branches, showing attractors are robust across a broad class of higher-curvature theories. These results underscore the infrared robustness of black hole entropy and scalar fixed points in settings relevant to string-inspired corrections.

Abstract

We investigate the attractor mechanism for spherically symmetric extremal black holes in a theory of general $R^2$ gravity in 4-dimensions, coupled to gauge fields and moduli fields. For the general $R^2$ theory, we look for solutions which are analytic near the horizon, show that they exist and enjoy the attractor behavior. The attractor point is determined by extremization of an effective potential at the horizon. This analysis includes the backreaction and supports the validity of non-supersymmetric attractors in the presence of higher derivative interactions. To include a wider class of solutions, we continue our analysis for the specific case of a Gauss-Bonnet theory which is non-topological, due to the coupling of Gauss-Bonnet terms to the moduli fields. We find that the regularity of moduli fields at the horizon is sufficient for attractor behavior. For the non-analytic sector, this regularity condition in turns implies the minimality of the effective potential at the attractor point.

Figures (2)

• Figure 1: $\phi(r)$ vs. $\log(r)$, where numerical coefficients are $\eta=1, \beta=2, \delta=3, \mu=5$ and $g_0=4$. Different curves represent different asymptotic values for $K=\phi_{\infty}$. The attractor point is $\phi_0=1$ at the horizon, $r_H=1$.
• Figure 2: $\phi(r)$ vs. $\log(r)$, where numerical coefficients for the potentials are $g_2=v_{i\ge2}=1$ and $g_1=2$. Different curves represent different asymptotic values for $\phi_{\infty}$. The attractor point is $\phi_0=0$ at the horizon, $r_H=1$.